Instability and self-contact phenomena in the writhing of clamped rods

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International Journal of Mechanical Sciences 45 (2003) 161 – 196
Instability and self-contact phenomena in the
writhing of clamped rods
G.H.M. van der Heijden∗ , S. Neukirch, V.G.A. Goss, J.M.T. Thompson
Centre for Nonlinear Dynamics, University College London, Gower Street, London WC1E 6BT, UK
Received 25 October 2001; accepted 24 November 2002
Abstract
We use the Cosserat rod theory to present a uni5ed picture of jump phenomena, associated with looping,
snap-through, pop-out, etc., in twisted clamped rods undergoing large de6ections. Both contact-free rods and
rods with isolated points of self-contact are considered. Taking proper account of the symmetries of the
problem we 5nd that an arbitrary contact-free solution is fully characterised by four parameters; each point
contact adds another two. A shooting method is used for solving the boundary value problem. An intricate
bifurcation picture emerges with a strong interplay between planar and spatial rod con5gurations. We 5nd new
jump phenomena by treating the ratio of torsional to bending sti8ness of the rod as a bifurcation parameter.
Load-de6ection curves are computed and compared with results from carefully conducted experiments on
contact-free as well as self-contacting metal-alloy rods.
? 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Cosserat rod; Clamped boundary conditions; End rotation, (out-of-plane) buckling; Instability; Bifurcation;
Self-contact; Looping; Snap-through; Pop-out; Experiments
1. Introduction
We study the global behaviour of clamped rods, that is, rods whose ends are 5xed into parallel
end-5ttings that can be moved inwards towards each other and rotated (see Fig. 1). We especially focus on dynamic jump phenomena that may occur when the end-to-end displacement or the end-to-end
rotation is varied. Examples of such jump phenomena are looping, snap-through, pop-out, etc., where
by ‘looping’ we mean a jump of a twisted rod into self-contact, ‘snap-through’ is used for a jump
not leading to self-contact, and ‘pop-out’ is used for a jump out of self-contact. All these phenomena,
and their associated hysteresis cycles, are readily observed in simple hand-held experiments, and we
∗
Corresponding author. Tel.: +44-(0)20-7679-2727; fax: +44-(0)20-7380-0986.
E-mail address: [email protected] (G.H.M. van der Heijden).
0020-7403/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0020-7403(02)00183-2
162
G.H.M. van der Heijden et al. / International Journal of Mechanical Sciences 45 (2003) 161 – 196
Fig. 1. The testing rig showing the force transducer on the right and the torque transducer on the left of the clamps
holding the rod.
encourage the interested reader to keep a piece of rod within reach (a silicone rubber specimen will
do nicely). More sophisticated laboratory tests with metal-alloy rods (see Fig. 1) have been found
to give results, to be presented later, which are also in reasonable quantitative agreement with the
theory. Mathematically, most of these jump phenomena can be characterised as fold or pitchfork
bifurcations.
Clamped end conditions tend to impose stronger constraints on the rod’s con5guration in space
and to produce more stable solutions and a richer bifurcation set than for instance found in a rod held
in a sleeve (so the end is free to rotate about its axis, and no twisting moment can be transmitted),
a rod supported by a Cardan (or Hookean, or universal) joint (so the end is free to rotate about
two orthogonal axes, one 5xed in the rod and one 5xed in space), or in a rod supported by a
ball-and-socket joint (so the end is free to rotate about three independent axes, and the total moment
is zero at the end). Clamping is also the natural way of supporting a rod in many engineering
applications.
Jump phenomena in end-loaded rods have been studied in scattered places in the literature. On
the one hand there are studies of looping and pop-out of twisted cables (e.g., References [1–3]),
or compressed rods (e.g., References [4–6]). On the other hand there are more mathematical works
concerned mainly with the existence of points of instability (e.g., References [7,8]). In most of these
studies, however, only one particular phenomenon is considered ([5,8] are more inclusive). Here we
present a more complete picture of how pre- and post-jump solutions are connected in the solution
space. Along the way we derive several exact analytical expressions for critical buckling loads not
seen previously in the literature. New jump phenomena are found by exploring a range of values for
the ratio of torsional to bending sti8ness of the rod. We treat both free rods and rods with one or multiple points of self-contact for which we model the rod as an impenetrable elastic tube of 5nite radius.
Our starting point is formed by the Kirchho8 equations for a perfect, inextensible, isotropic rod
undergoing arbitrary deformations under the action of end loads. A singularity-free (in fact, polynomial) ODE is obtained by avoiding the usual Euler angles and working with the position and
tangent vectors as variables instead. Although the equations for a perfect, isotropic rod are known
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to be solvable in terms of elliptic integrals and functions, this is not particularly helpful if one
is interested in numerical results, especially when considering rigid, i.e., displacement-controlled,
loading, as the implicit transcendental equations have to be solved numerically. Also, working with
closed-form solutions leads to annoying discontinuities when parameters are varied. Therefore we
use numerical solution methods throughout. The results give a coherent and uni5ed picture of the
buckling and post-buckling of relatively short pieces of rod and allow full loading sequences to be
followed and hysteresis loops to be identi5ed. Since most of the instabilities encountered pose a
threat to the proper operation of pipes and cables in industrial environments, this approach may be
useful in designing operating conditions with minimum risk of looping or snap-through, or conditions
which favour pop-out in order to avoid kinking (i.e., permanent plastic deformation).
We mostly study open rods as these are more conventionally found in engineering applications.
Nevertheless, the planar ring is found to be an important organising centre for deformations of
both the planar and the spatial elastica. The ring is an example of a closed rod and for such rods
we can de5ne the so-called link as the topological linking number of two lines drawn on opposite
sides of the unstressed rod [9]. This link is a topological invariant, i.e., it is una8ected by continuous
deformations. This and the fact that the (clamped!) boundary conditions for a closed rod are naturally
prescribed make closed rods an attractive object for studies from a more mathematical viewpoint.
Indeed, their symmetry and bifurcation properties have been studied extensively by Maddocks and
co-workers (e.g., Reference [10]). Closed rods with self-contact have more recently been studied by
Coleman and co-workers [11,12] as models for DNA plasmids.
Clamped rods have several attractive features. Firstly, as we shall show, clamped boundary conditions applied to the (reversible) equilibrium equations imply that all open-rod solutions have re6ection
(anti-)symmetry about the midpoint. So one need only compute half solutions; full solutions follow
by suitable re6ection. Secondly, because in clamped rods the end tangents are aligned, the natural
end displacements most readily accessible in an experiment, namely the end shortening (or slack)
and the end rotation, are also the ones through which the end tension and twisting moment do work.
Thus, by using end shortening and end rotation in load-de6ection diagrams one can infer static
stability properties of solutions by making use of the so-called fold rule [13,14]. The end rotation,
in a sense to be made precise, can be seen as a generalisation of the link, well-de5ned for closed
rods, to arbitrary clamped con5gurations. Although an important quantity, it has not been considered
in the literature so much ([5] is a notable exception).
For self-contacting con5gurations we 5nd a bifurcation sequence in which the rod jumps to states
of increasing number of self-contacts as a control parameter (end rotation or end shortening) is
increased, resulting in a ply structure known, for instance, from DNA supercoiling [12,15] and textile
yarn twisting [16]. This process of ply formation, although remarkably delicate in its theoretical detail,
can easily be observed in a rubber rod. The beginning of ply formation can be seen in Fig. 12.
The organisation of the paper is as follows. In Section 2 we formulate the problem of a clamped
contact-free rod. We introduce the end rotation as a measure of torsional de6ection and discuss its
relationship to link and writhe de5ned for closed rods. Taking into account the symmetries of the
problem we show that for a given rod a clamped solution is fully characterised by four parameters.
In Section 3 we present our numerical results focussing on the various jump phenomena found.
We also use the fold rule to ascertain the (in)stability of solution branches. Section 4 sets up the
equations for a rod with an arbitrary number of isolated point contacts. For a symmetric solution
each point contact requires an additional two parameters to be speci5ed. Numerical results, in the
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Table 1
List of frequently used symbols
L; B; C; F; F; f
T; t
NF
M; m
M0 ; m0 ; m0x ; mz
M
D; d
R
Length and dimensionless radius of the rod
Bending and torsional sti8ness, and sti8ness ratio C=B
(dimensionless) Internal force
(dimensionless) End tension
Magnitude of point contact force
(dimensionless) Internal moment
(dimensionless) Reference moments at the midpoint of the rod
End twisting moment
(dimensionless) End shortening
End rotation
form of load-de6ection curves, for self-contacting rods
some results from our experiments, and Section 7 closes
we spell out a proof that deformations of clamped rods
For easy reference, Table 1 gives a list of frequently
are given in Section 5. Section 6 presents
this study with a discussion. In an appendix
are symmetric.
used symbols.
2. Formulation of a clamped rod
2.1. Equilibrium equations
Consider a thin elastic rod of length L held by end forces and moments. If we denote the position
of the rod’s centreline by the vector function R, then the internal force, F, and moment, M , in an
element of the rod are governed by the balance equations [17]
F = 0;
(1)
M + R × F = 0;
(2)
( )
= d=dS, S being the arclength parameter measured along the centreline of the rod. We
where
shall take the origin of arclength at the middle of the rod, so that −L=2 6 S 6 L=2. We assume
the rod to be perfect (i.e., intrinsically straight and untwisted), inextensible and unshearable. Thus
the only distortions which the rod su8ers are on account of bending and twisting moments, and not
of shear forces or tension. Let {d1 ; d2 ; d3 } be a right-handed rod-centred orthonormal co-ordinate
frame (the directors) with d3 the local tangent to the rod and d1 and d2 two vectors in the normal
cross-section coincident with the principal bending axes. Thus we have
R = d3 ;
(3)
while the evolution of the co-ordinate frame along the rod is governed by the equation
di = u × di
(i = 1; 2; 3):
(4)
Here u is the generalised strain vector whose components in the moving frame are the curvatures
and the twist. We introduce linear constitutive relations between the moments and the strains:
M · d 1 = B 1 u1 ;
M · d 2 = B 2 u2 ;
M · d3 = Cu3 :
(5)
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B1 and B2 are the principal bending sti8nesses about d1 and d2 , respectively, while C is the torsional
sti8ness about d3 .
Clearly, F is a constant vector in space. Assuming this vector to be non-zero we can de5ne a
5xed orthonormal co-ordinate frame {ex ; ey ; ez } with ez taken in the direction of the internal force:
F = Fez ;
F ¿ 0;
(6)
and ex and ey such that at S = 0 the tangent to the rod lies in the (ex ; ez ) plane (i.e., d3 (0) · ey = 0).
This can be done without loss of generality. (The force F will only be zero at isolated points along
solution curves to be presented in the following sections. These points correspond to straight rods
or twistless rings, and since these can be dealt with separately (see Table 2), the above de5nition
gives no problems.) Eq. (2) can now also be integrated to give
M = Fez × R + M0 ;
(7)
where M0 is an integration constant which has to be determined through the boundary conditions
as part of the problem. It easily follows that the equilibrium equations have the two 5rst integrals
(constants or conserved quantities, not varying with S)
M · F = FM · ez
and
1
M
2
· u + F · d3 = 12 (B1 u12 + B2 u22 + Cu32 ) + Fd3 · ez :
(8)
The 5rst of these, after division by F, is the moment about the internal force vector (‘the wrench
moment’), while the second, interpreted properly, is the Hamiltonian [18]. The fact that the resultant
of the vector force and vector moment at any cross-section is statically equivalent to a wrench 5xed
in space is an important concept. As we vary the loading of the rod, and move along an equilibrium
path, the orientation of the wrench axis will normally vary, typically becoming displaced and rotated
relative to the 5xed axis of the loading rig. We shall return to this in Section 5.2.
In this paper we shall only be concerned with an isotropic rod, i.e., one for which B1 = B2 = B,
say. Then the twisting moment M · d3 is a further 5rst integral and we have Bd3 = M × d3 . Therefore
Bd3 = (Fez × R + M0 ) × d3 :
(9)
As we shall be interested in rigid loading sequences in which the end rotation of the rod is controlled
we must follow the evolution along the rod of a material vector such as d1 . Noting that
d3 × d3 = u − u3 d3 ;
(10)
we can use (4) to obtain
Bd1 = (Fez × R + M0 ) × d1 + (B − C)u3 d3 × d1 ;
(11)
where u3 is the (constant) twist.
Thus we have arrived at a system of three Eqs. (3), (9) and (11) for the vector functions R, d3
and d1 , with parameters F, M0 and u3 . The equations satisfy the constraints
d3 · d3 = 1;
d1 · d1 = 1;
d1 · d3 = 0:
(12)
Since the left-hand sides in (12) are 5rst integrals of the equations, these constraints can be ensured
by suitable boundary conditions.
166
2.2. Symmetries and boundary conditions
For reasons explained in the Introduction we consider clamped boundary conditions, which can
be formulated as
d3 (−L=2) = d3 (L=2);
(13)
R(L=2) − R(−L=2) = d3 (L=2)
for some ∈ R:
(14)
Now the rod equations are well-known to have (reversing) symmetry properties [19]. These symmetries imply that solutions are either themselves reversible or come in reversible pairs. In our case of
clamped rods one can prove that solutions selected by the boundary conditions are in fact reversible.
Since the proof is most easily given using Euler angles we defer it to the Appendix. A consequence
of the symmetry is that M · ey is an odd function of arclength, so provided we take the initial
conditions
R(0) = (X (0); Y (0); Z(0)) = (0; 0; 0);
(15)
so that by (7) we have M0 = M (0) = (M0x ; 0; M0z ), the reversing symmetry manifests itself as
invariance of the equations (3), (9) and (11) under the following transformation:
S → −S;
X → −X;
Y → Y;
Z → −Z;
d3x → d3x ;
d3y → −d3y ;
d3z → d3z ;
d1x → d1x ;
d1y → −d1y ;
d1z → d1z ;
(16)
where we have introduced components of vectors with respect to the 5xed co-ordinate frame
{ex ; ey ; ez }. (Since Mz (s) is constant, we shall henceforth refer to M0z simply as Mz .)
This invariance implies that we can restrict ourselves to the arclength interval [0; L=2], and specify
boundary conditions at S = 0 and S = L=2. Then (13) and (14) reduce to two clamp conditions,
d3y (L=2) = 0;
X (L=2)d3z (L=2) = Z(L=2)d3x (L=2);
(17)
and a formula for the span length ,
= 2{X (L=2)d3x (L=2) + Z(L=2)d3z (L=2)}:
(18)
Further, since d3 is a unit vector and d3y (0) = 0 we can specify
d3 (0) = (d3x (0); d3y (0); d3z (0)) = (sin ; 0; cos );
(19)
where ∈ [0; 2) is the angle the rod at the midpoint makes with ez , and
d1 (0) = (cos ; 0; −sin ):
(20)
Note that this choice satis5es the constraints (12).
Since F (the ‘wrench force’) by de5nition is positive, the sign of the tangent vector d3 determines
whether the rod is locally in tension or compression. In particular, the component of the end force
in the direction of the rig axis (the ‘rig force’) is given by
T = F · d3 (L=2) = Fd3z (L=2);
(21)
167
positive for tension, negative for compression. Meanwhile, the end twisting moment (or ‘rig moment’) is, from (5),
M = M · d3 (L=2) = Cu3 :
(22)
These T and M will be used to present our results since they are the quantities most readily measured
in an experiment.
The end shortening, or slack, is de5ned as
D = L − = L − 2{X (L=2)d3x (L=2) + Z(L=2)d3z (L=2)}:
(23)
It measures the di8erence in the end-to-end distance of the buckled and the straight rod. Note that
D takes on values between 0 and 2L, and that D = L for a closed rod. For D ¿ L the rod leaves
the clamps on the outside (see, e.g., Fig. 2(a)). We shall call such solutions inverted. Notice that
we are here ignoring the self-intersection at the clamps that would arise in a loading sequence in
which D passed through the value L.
Since the end tangents are aligned (by (13)) we can de5ne the unique reduced end rotation
R̂ ∈ [0; 2) by
cos R̂ = d1 (L=2) · d1 (−L=2);
sin R̂ = d1 (L=2) · d2 (−L=2);
(24)
where d2 = d3 × d1 . The full end rotation R can be written as R = R̂ + 2k, where k is an integer
that can be determined by using continuity. More precisely, we de5ne the end rotation of a clamped
rod in arbitrary deformation to be the end rotation that rod has ‘when completely pulled straight’
without (further) rotation of the ends (i.e., in the limit D → 0, where simply R = ML=C). For this
de5nition to be applicable any arbitrary clamped solution must be connected, by a continuous path
of 3D solutions, to a unique (twisted) straight solution. Although we do not have a proof of this
it is our 5rm conjecture, based on experience in our extensive numerical computations, that this is
indeed the case, i.e., there appear to be no isolated (either open or closed) branches of solutions.
This means that numerical continuation, in the right direction, can be employed to ‘trace back’ a
given solution to D = 0 where k (or R) can be read o8 immediately.
The (numerical) end rotation R as de5ned above has the advantages that it is well de5ned for any
arbitrary con5guration and that it permits us to distinguish between di8erent con5gurations that have
the same reduced end rotation R̂. However, it has to be realised that during parameter continuation
in, for instance, D (as in Fig. 3), the rod con5guration may go through a self-intersection (which
would not a8ect R). In consequence, R does not necessarily agree with the physical end rotation
(which a real rod would have that could not intersect itself). The problem with this physical end
rotation, however, is that it is not straightforward to de5ne generally.
One approach would be to ‘pull the rod straight’ by monotonically decreasing D without (further)
rotation of the ends. In this process, which mimics the actual straightening of an arbitrarily deformed
physical rod, the rod would be allowed to jump dynamically to another solution branch (possibly one
of self-contacting solutions, as computed in Section 5) when a limit point is encountered. This new
branch may be disconnected from the initial branch (if the mode of the solution is di8erent). The
problem is that if multiple stable branches are available, then it is not a priori clear to which branch
the rod will jump. This would require a dynamical theory. Even worse, if, along the loading path,
the solution has gone through a self-intersection, then the rod may end up in a knotted con5guration,
in which case D = 0 cannot be reached (for a rod of 5nite thickness) and the approach to de5ning
a physical end rotation fails utterly.
168
A more promising route would be to generalise the link, a topological quantity de5ned for a
closed rod, to open rods. For a closed rod the link, Lk, is given by the linking number of the two
curves r + !d1 and r − !d1 (which can be thought of as two curves drawn on opposite sides of the
rod). For suQciently small ! this is well de5ned and we have the celebrated formula [9,20]
Lk = Tw + Wr:
(25)
Here Tw = (1=2) L u3 dS is the total twist, and Wr the writhe. The writhe is a property of the
centreline of the rod and is a measure of the out-of-plane deformation (it is not a topological
invariant). It can be computed as the double integral
[R (s) × R (s )] · [R(s) − R(s )]
1
ds ds ;
(26)
Wr =
4 L L
|R(s) − R(s )|3
and is also equal to the signed crossing number averaged over planar projections from all possible
directions [9]. An elementary exposition of link and writhe can be found in Reference [21].
The de5nitions of link and writhe can be applied to open rods by imagining a closing piece of
material connecting the ends of the deforming rod (such a closure was introduced earlier in Reference
[22]). This closure can be assumed to be twistless (by imagining it to be in5nitely sti8 in torsion)
so that all the twist of the closed structure is in the rod. Then the total structure (rod plus closure)
is closed and Lk and Wr are well-de5ned. They will however depend on the shape of the closure.
For instance, if the rod is bent into a twistless circle then the closure will also form a closed curve
and the total structure has a self-crossing. If both loops of this self-crossing structure are in the
same plane then the writhe of the total structure will be +1 (for a suitable sign convention) and, Tw
being 0, by (25), Lk = 1. Since for a twistless circular rod R = 2, we conclude that Lk = R=(2),
i.e., the end rotation is essentially the link of the closed structure. For this relationship to hold
generally, self-intersections in either the rod or the closure must be disallowed during any given
loading sequence. We have already concluded that self-intersections do not a8ect the numerical end
rotation R, but the writhe Wr and the link Lk will jump by ±2. (In case two (or any even number
n of) symmetrically placed self-intersections occur simultaneously, then the total change in Wr and
Lk will be ±2n.)
This shows the signi5cant problems with trying to generalise the link to open rods and to relate
it to the physical end rotation, i.e., the measure of torsional de6ection that is relevant for energy
purposes. The matter will be pursued in detail elsewhere. In the remainder of this paper we shall
only use the numerical end rotation R as de5ned above. As long as no self-intersections occur during
a loading sequence this R agrees with the physical end rotation.
2.3. Solving the equations
Throughout this paper we present results in dimensionless form by using the length L and the
bending sti8ness B as characteristic scales. Occasionally we shall 5nd it convenient to use the
following abbreviations:
S
R
D
FL2
s= ; r= ; d= ; f= 2 ;
L
L
L
4 B
t=
TL2
;
42 B
m=
ML
;
2B
=
C
;
B
(27)
169
and to write r = (x; y; z) and m(0) = m0 = (m0x ; m0y ; mz ). We then have a system of nine equations
(3), (9), (11) with nine initial conditions (15), (19), (20), a solution of which is fully speci5ed by
the constant (5xed for a given rod) and the four dimensionless parameters
m0x ;
mz ;
f;
;
(28)
from which the dimensionless twist can be computed as
Lu3 1
= (m0x sin + mz cos ):
2
(29)
The end displacements d and R are hereby determined through (23) and (24). The parameters (28)
are found by applying the boundary conditions. In a rigid-load situation the end displacements d
and R will be prescribed, and the two clamp conditions (17) together with (23) and one of Eqs.
in (24) provide a well-posed problem of four equations for the four unknowns (28). In a typical
loading sequence d (or R) is then varied and a curve of solutions is traced out.
It should be remarked that Eqs. (3) and (9), with the 5rst integrals as mentioned in Section 2.1,
are completely integrable and solvable by quadrature. The integrations are most conveniently carried
out in a cylindrical co-ordinate system parallel to ez [18] and yield expressions in terms of elliptic
integrals and functions (see, e.g., Reference [23]). However, in the displacement-controlled loading
case, as opposed to the dead (i.e., traction-controlled) loading case, m0 and f are unknown and one
ends up with a set of implicit transcendental equations which have to be solved numerically. (This
is even more so in the contact problem considered in Section 4.) Therefore we shall use a numerical
approach throughout (although some analytical expressions for planar solutions and critical parameter
values will be given in Section 3).
Thus the 5nal boundary value problem is solved over the interval [0; 1=2] using a shooting method
in which the parameters (28) are varied. The other half of the solution is obtained by suitable
re6ection (cf. (16)). For the numerical integration we use the highly accurate 8th-order Runge-Kutta
code DOP853 as described in Reference [24], with local error tolerance 10−12 . For single-parameter
continuation of solutions we use the code AUTO [25] which uses orthogonal collocation allied to
pseudo-arclength continuation. Eq. (11) is slaved to (3) and (9), and is only integrated in case one
requires (control of) the end rotation.
Special solutions, such as planar ones, are obtained for speci5c parameter values. Table 2 gives
a summary. A more extensive classi5cation of 2D as well as 3D clamped rod con5gurations will
be given in Reference [26]. As an example we now demonstrate that mz = 0 = u3 implies a planar
solution (the converse is not true). From (29) it follows that there are two cases: (i) = 0; (case
6 in Table 2) and (ii) m0x = 0 (case 7 in Table 2). In case (i) d3x (0) = 0 by (19), and we have the
equations d3x = my d3z , my = fd3x . Since my (0) = 0 by reversibility, it follows that all the derivatives
of d3x are zero at s = 0, and hence that d3x ≡ 0 (and thus my ≡ 0). Since x(0) = 0 this means that
x ≡ 0 and the solution lies in the y–z plane. If we introduce the angle & such that d3y = sin &,
d3z = cos &, then the remaining equations d3z = mx d3y , mx = −fd3y yield the familiar equations for
the Euler elastica, & − f sin & = 0, mx = −& . In case (ii) we have the equations d3y = −mx d3z ,
mx = −fd3y , and since d3y (0) = 0 we similarly 5nd that d3y ≡ 0 (and thus mx ≡ 0). This, with
y(0) = 0, gives y ≡ 0, so the solution is seen to lie in the x–z plane.
170
Table 2
Summary of special solutions for a clamped rod (∗ means any value)
m0x
1
2
mz
0
f
Type of solution
0
0
Straight and prismatic, unstressed
Untwisted n-covered planar ring
(M = const:)
Straight and prismatic,
in tension ( = 0) or compression ( = )
Straight and twisted,
in tension ( = 0) or compression ( = )
Twisted n-covered planar ring
Odd-mode in6ectional ( = ) or
any-mode non-in6ectional ( = 0 or )
elastica in y–z plane
Even-mode in6ectional elastica in x–z plane
Untwisted 3D solution
3D solution with M · F = 0
3
0
= 0
m20x + m2z = n2
0
0
= 0
∗
tan = −mz =m0x
(limit =2)
0; 4
0
= 0
= 0
0; 5
6
±nF
= 0
±n
0
= 0
= 0
=2; 3=2
0; 7
8
9
0
= 0
= 0
0
= 0
0
= 0
= 0
= 0
∗
tan = −mz =m0x
∗
3. Results of numerical continuation
Initially we take the value 5/7 for the sti8ness ratio C=B. This is the value for the nitinol rods
which we used in experiments (see Section 6). According to the theory of linear elasticity for rods
of solid homogeneous circular cross-section, as the ones we used, this corresponds to a Poisson ratio
' = B=C − 1 = 0:4. Later C=B will be varied.
Where possible we use the fold rule to infer (static) stability properties of the solution branches
computed. Standard bifurcation theory [27] predicts that folds in bifurcation diagrams mark exchanges
of stability. From elastic stability theory it is well-known that in diagrams in which pairs of conjugate
variables (one of which doing work through the other) are plotted, one can not only infer a change
of stability but also the direction of change as a solution curve goes through a fold [13]. Examples
of such distinguished diagrams in the present context are load-de6ection diagrams of rig tension T
against end shortening D and rig moment M against end rotation R.
This fold rule is an instance of the application of the Morse index as de5ned for a certain class
of boundary value problems having an underlying variational structure [14]. The index gives the
maximum dimension of the space on which the second variation can be made negative. Provided
some genericity conditions are satis5ed the index changes by one each time a fold is traversed, and
a solution can be stable only if it has index zero. Since only changes of index are recorded one
needs stability information of one solution branch by other means in order to start the method o8.
Once that information is available (for instance, by physical means) the shape of solution branches
in distinguished diagrams determines which solutions have non-zero index, and which therefore are
unstable. Recent extensions of the use of an index (for instance to constrained problems and to cases
where bifurcations occur) can be found in References [28–30].
In the present case the total potential energy V in the rod is given by [22]:
1
V=
(Bu12 + Bu22 + Cu32 ) dS + TD − MR:
(30)
2
171
It then follows that in D–T diagrams presented below the lower branch of solutions in a fold opening
to the right, and the upper branch in a fold opening to the left, cannot represent local minima of the
functional V considered as a function of the variable T and bifurcation parameter D. More generally,
the index in the former case is decreased by 1, and in the latter case increased by 1, as one traverses
the fold in upward direction (at constant R). Because of the di8erent signs of the work terms in
(30) the situation is the opposite in the R–M diagrams presented below: in a fold opening to the
right the upper branch is unstable, while in a fold opening to the left the lower branch is unstable.
These rules are applied in Figs. 3 and 7 below.
3.1. Primary buckling
A clamped straight non-twisted rod buckles into a planar in6ectional Euler elastica at the critical
load t =TL2 =(42 B)=−1. The load–de6ection curve for a general in6ectional n-mode (i.e., a clamped
planar elastica with n + 1 in6ection points) for n odd is given parametrically in terms of the elliptic
modulus k (0 6 k 6 1) by the expressions (see, e.g., [31])
E(k)
TL2
D
2 2
= −4(n + 1) K (k);
=2 1−
;
(31)
B
L
K(k)
where D is the end shortening and K(k) and E(k) are the Legendre complete elliptic integrals of
the 5rst and second kind, respectively. For n even no such explicit parametrisation exists, but one
can solve the equation for the planar elastica and apply the boundary conditions (13) and (14) to
obtain a set of transcendental equations. For a uniform rod all buckling modes but the primary one
(n = 1) are unstable, both in the plane and spatially, as 5rst shown in Reference [8].
For an n-mode (any n) non-in6ectional clamped elastica (i.e., one with n loops) the expressions
are:
TL2
D
E(k)
D
2
2 2 2
= 4n k K (k);
=2− 2 1−
¡1 ;
B
L
k
K(k)
L
(32)
2
D
E(k)
D
TL2
= −4n2 k 2 K 2 (k);
= 2 1−
¿1 :
B
L
k
K(k)
L
Fig. 2(a) shows curves for the 5rst four in6ectional as well as non-in6ectional modes. Note that the
‘5gure-of-eight’ appears on both the 5rst- and second-mode in6ectional branch. On the former it is
held at the edge and requires a compression, while on the latter it is held at the centre and requires
a tension. (Despite the super5cial similarity this 5gure-of-eight is not Bernoulli’s lemniscate.) In
Reference [32] a branch of (non-reversible) solutions is computed connecting the two 5gures-of-eight.
A non-trivial planar elastica (i.e., not a straight rod or a ring) always has mz =0=u3 (cf. Table 2).
From the scaling in (27), and (31) and (32), it follows that if the parameters (m0x ; t; ) give
a 5rst-mode non-trivial planar elastica with end shortening d, then (nm0x ; n2 t; ) give an n-mode
non-trivial planar elastica with end shortening d for an odd-mode in6ectional or an odd-mode
non-in6ectional elastica, and end shortening 2 − d for an even-mode non-in6ectional elastica. The
scaling behaviour of the even-mode in6ectional elastica, for which the rig and wrench axes make
an angle (see the shapes in the insets of Fig. 2), is less straightforward.
172
6
10
1
2
3
4
4
0
ML/2πB
TL2/4π2B
5
1
2
-5
0
-2
3
-4
4
-6
-10
0
(a)
2
0.5
1
D/L
1.5
-6
2
(b)
-4
-2
0
2
TL2/4π2B
Fig. 2. (a) Labelled solution branches for the 5rst four in6ectional and non-in6ectional modes of the planar elastica
as given by (31) and (32). The in6ectional ones bifurcate from the straight rod (vertical axis) and have R = 0, the
non-in6ectional ones all pass through the ring at D=L = 1 (marked by the solid circle) and have R = 2n, where n (n ¿ 1)
is the mode number. Insets show true-view, and 5xed-length, shapes of the 5rst and second mode; and (b) buckling curves
for the 5rst four modes of a clamped twisted rod as given by (33). Insets of the 5rst and second mode of the planar
elastica, at M = 0, illustrate the di8erence between odd and even modes, with the dotted lines indicating the wrench axes.
If a clamped straight rod is subject to both compression and twist then it buckles at a critical load
given by
cos m2z − 4t − cos mz = (2t sin m2z − 4t)= m2z − 4t;
(33)
as follows from a linearisation of Eqs. (3) and (9) about the trivial solution. This condition de5nes
a countable in5nity of disjoint buckling curves in the t-mz load plane, the 5rst four of which are
drawn in Fig. 2(b). We are not aware of the appearance of (33) anywhere in classical works such
as References [31,33], although the latter deals with the special case of pure torsional buckling
(i.e., t = 0), when (33) reduces to tan mz = mz . Condition (33) is, however, derived in Reference
[34], where it is mentioned that the result agrees with results in References [35,36]. (Greenhill [37],
who gave the correct condition m2z = 1 + 4t for pinned ends, guessed the formula m2z = 4 + 4t for
the 5rst clamped buckling mode, but this is incorrect except for pure-thrust buckling for which
mz = 0. Incidentally, pinned boundary conditions for a twisted rod in space are here taken to mean:
X (−L=2) = 0 = X (L=2); Y (−L=2) = 0 = Y (L=2); Mx (−L=2) = 0 = Mx (L=2); My (−L=2) = 0 = My (L=2).)
Although, as in the planar case, higher-order modes are unstable [30] and will not be observed near
buckling, their solution curves may go through the requisite number of folds to produce stable solutions. For instance, Figs. 3 and 6 show second-mode curves which give rise to stable branches going
through the ring. Also, di8erent modes interact with each other away from buckling, so they have
to be considered if one is to get a global picture of the solution set. Indeed, it is not straightforward
to de5ne the ‘mode’ of a solution away from the linear regime near primary buckling. However,
since a given solution can be traced back to a unique nearly straight solution near buckling we can
de5ne the mode of a solution by reference to this ‘linear’ solution. Although not always satisfactory
(for instance, it makes the mode of the planar ring unde5ned), this de5nition is suQcient for our
present purposes. A more careful de5nition is given in Reference [26] by using the representation
173
4
γ = 5/7
3
TL2/4π2B
2
1
(e)
(d)
(c)
(b)
(a)
0
-1
-2
(a)
(f)
(c)
(b)
(a)
-3
-4
(f)
(f)
-5
0
0.5
1
1.5
2
D/L
Fig. 3. Load–de6ection diagram with solution curves for R = 0 (a), (b), 2 (c), 3 (d), 4 (e) and R = Rc = 8:9527
(f). Solid lines represent stable branches, dashed lines unstable branches under rigid loading. Short-dashed curves are for
the planar elastica, either looped and non-in6ectional (upper curve), or in6ectional (lower curve, retraced by the R = 0
curves, but emanating from the Euler compressive buckling load −1). The triangle indicates the out-of-plane instability
for R = 0 at D=L = 0:5590, the diamond the into-the-plane instability for R = 2 at D=L = 0:6767. The solid circle marks
the ring. ( = 5=7).
in terms of elliptic integrals and functions. It is then found that the mode changes when and only
when D=L goes through one (again rendering the mode of the planar ring ambiguous).
3.2. Out-of-plane buckling and beyond
Fig. 3 shows D–T solution curves for several values of R. Two di8erent sets of curves can be
distinguished: those that have R ¿ 2, emerge from 5rst-order modes at buckling (on the vertical
axis), then go through a fold, and end with T → ∞, and those that go through the planar ring
(marked by the solid circle). The former will be further discussed in the next section; the latter,
which show more complicated behaviour, form the topic of this section.
To start with the planar solutions, we observe that end shortening of the planar rod (R = 0)
initially takes place under a rising compressive load. At a critical compression −T and end shortening
D, given by a critical modulus kout in (31), the planar rod becomes unstable and buckles out of
the plane. The critical point, described by a pitchfork bifurcation, is indicated by the triangle in
Fig. 3. At this point the axial compressive load starts to drop with increasing end shortening. The
load becomes slightly tensile for a brief period before the shape of the rod reaches the circular ring
with one turn of twist. Upon further increase of D the con5guration becomes inverted and slightly
non-planar (it was shown in Reference [38] that the only planar con5gurations of a twisted rod that
can be supported by end loads only are the straight line and the circle).
The location of the out-of-plane bifurcation can be calculated analytically by linearising the 3D
equilibrium equations about the planar in6ectional elastica and solving the resulting equations subject
to our clamped boundary conditions to derive a condition for the existence of neighbouring non-planar
solutions. The calculations are most conveniently carried out using Euler angles and involve the
evaluation of numerous integrals of combinations of Jacobian elliptic functions. We omit the details
D/L
174
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0
0.5
1
1.5
2
2.5
3
3.5
4
C/B
Fig. 4. Loci of the out-of-plane (solid) and into-the-plane (dashed) bifurcations occurring along the R = 0 and R = 2
curves, respectively. Insets show true views of the rod’s shape at several of the points of lateral instability. The interval
2=3 ¡ C=B ¡ 1 (0 ¡ ' ¡ 1=2) delimited by the diamonds is the region of prime, though not sole, physical interest.
here and refer to Zajac [1] for a similar calculation for the non-in6ectional elastica (a result we shall
discuss shortly). The result can be presented in the following parametrised form:
E(k)
C
K(k)[2(1 − k 2 )K(k) + (−3 + 4k 2 )E(k)]
D
=2 1−
;
=
:
(34)
L
K(k)
B
2(1 − k 2 )K 2 (k) + (−5 + 4k 2 )K(k)E(k) + 3E 2 (k)
The corresponding critical buckling load (for an arbitrary odd mode!) is obtained from (31).
Although this out-of-plane buckling instability has a sizable literature, (34) appears to be new,
although Miyazaki and Kondo [5] give a highly implicit expression for the dependence of the critical
load on the torsional sti8ness. Previous studies of this secondary bifurcation further include: KovTari
[4], who considers the case of an isotropic rod free to twist at one end, Antman [7], who studies
the linearisation of the non-linear equations about a planar solution in order to obtain existence
results, and Maddocks [8], who considers bifurcation under several types of boundary conditions in
dead load using variational methods. The instability was studied in a perturbative approach allowing
extension to anisotropic and imperfect rods in Reference [6] in the context of birdcaging in wound
cables.
The solid curve in Fig. 4 gives a graphical representation of (34). Notice that for a large enough
ratio of torsional to bending sti8ness the onset of lateral instability may occur for con5gurations
beyond the 5gure-of-eight, i.e., for inverted solutions, a fact readily demonstrated with a vacuum
cleaner tube.
Moving on to the non-planar solutions we 5rst remark that for R somewhere between and 2
the primary bifurcation from the trivial state (on the vertical axis) changes from being supercritical
to being subcritical, with the bifurcating branch of solutions rising with increasing D. Although this
doesn’t mean much for a rod under rigid loading, for dead loading it means that the rod jumps,
rather than gradually evolves, out of its straight con5guration. This critical point was studied in
Reference [39] through a Lyapunov–Schmidt reduction for the slightly di8erent case of a rod with
prescribed parallel end tangents but with one end free to undergo side de6ection.
The curves for 0 ¡ R ¡ 2 that emerge from 5rst-order buckling modes are stable for all D. However, there are additional curves for the same R values which emanate from (unstable) higher-order
175
buckling modes, and undergo the right number and orientation of folds to have a stable branch of solutions through the
√ ring. This stable branch shrinks as R is increased and disappears
at R = Rc := 2( 3= − 1) at which point the planar ring becomes unstable (see curve (f) in
Fig. 3 with vertical tangent at the ring). None of the curves going through the ring having R ¿ Rc
appear to contain stable branches.
The critical point at Rc corresponds to the well-known twisted-ring instability criterion derived by
Zajac [1]. More generally, it can be shown [40] that the nth buckling mode (n ¿ 1) of the m-times
covered ring (m ¿ 1) bifurcates from the primary branch at a critical twisting moment given by
ML = (n + m)2 − m2 :
(35)
2B
(Apart from this unpublished work [40] to our knowledge this result is only quoted, albeit incorrectly,
in Reference [41].) As shown in Reference [28], this bifurcation is subcritical (with a jump into
self-contact) for small and supercritical (with a gradual deformation out of the plane) for large ,
the transition occurring at = c , where
3m2
:
(36)
c = 1 +
2(n + m)2
An early reference to stability studies of twisted elastic rings in the context of DNA is [42].
Notice that the curve for R = Rc in Fig. 3 ends in a third-order buckling mode at D = 0. What is
the highest mode with a stable branch
through the ring depends on . Numerically we 5nd this to
√
be the 2-mode for 0:803 ¡ ¡ 3 (consistent with Fig. 6) and the 3-mode for 0:11 ¡ ¡ 0:803.
An arbitrarily high mode may have a (small)
stable branch through the ring by taking suQciently
√
small. As follows from (35), for ¿ 3 the ring will be unstable even for R = 0, i.e., with one
full turn of twist put in, so no stable branches exist through the ring. Incidentally, returning to
the problems with de5ning a satisfactory end rotation discussed in Section 2.2, along the Rc curve
in Fig. 3 two simultaneous symmetric self-intersections of the rod axis occur somewhere between
D=L = 0:83 and D=L = 0:84.
The behaviour for even larger values of is depicted in Fig. 5 showing how the solution curve
for R = 0 gets more and more drawn into a spiralling structure around the ring point. Solution curves
for R between 0 and 2 follow this spiralling behaviour but with fewer and fewer oscillations
occurring as 2 is approached. Thus for these 5rst-mode solutions increasing R has a stabilising
e8ect (the converse is true for second-mode solutions). In view of the fold rule for stability change,
the spiralling does not create stable branches. Notice that the ring becomes unstable before the
5gure-of-eight gains stability but after the point where the planar elastica 5rst touches itself (at
k = 0:8551::: in (31)), so that the rod would have to intersect itself in order to follow the path
in Fig. 5(b) with the dynamic jump across the unstable circle. Also notice that at = 12:0 (Fig.
5(d)) the out-of-plane pitchfork bifurcation has become subcritical. The transition from supercritical
to subcritical occurs around = 9.
√
At 5rst sight the critical value = 3, let alone the higher values in Fig. 5, might appear to
be unrealistically high given that for solid circular rods of normal material the ratio of torsional
to bending sti8ness satis5es 2=3 ¡ ¡ 1 (corresponding to 0 ¡ ' ¡ 1=2). However, Kehrbaum and
Maddocks [43] have shown that a rod whose cross-section is anisotropic but which is rapidly twisting
in the unstressed state can be approximated by an e8ectively isotropic rod in which the ratio of the
e8ective torsional and bending sti8nesses can be much larger than 1. Also, direct experiments to
2
2
1
1
TL2/4π2B
TL2/4π2B
176
0
-1
0
-1
-2
-2
-3
-3
0
0.5
(a)
1
1.5
2
0
0.5
(b)
D/L
1
1.5
2
1.5
2
D/L
3
2
2
1
TL2/4π2B
TL2/4π2B
1
0
-1
0
-1
-2
-2
-3
-3
-4
0
(c)
0.5
1
1.5
D/L
2
0
(d)
0.5
1
D/L
Fig. 5. Solution curves for R=0 and varying . Shown are curves for =1:7 (a), 1.8 (b), 3.2 (c) and 12.0 (d). Short-dashed
curves are for the planar elastica. All curves are going
√ through the planar ring
√ at D=L=1 indicated by the solid circle. This
ring, with one full turn of twist, is stable for ¡ 3 and unstable for ¿ 3. The diamond indicates the 5gure-of-eight
for which −TL2 =42 B = 4K(k8 )2 =2 = 2:1833:::, where k8 = 0:9089::: solves K(k) = 2E(k), while the triangle marks the
self-touching elastica at k = 0:8551::: :
estimate the bending and torsional sti8nesses of DNA molecules have led to 0.7–1.5 as the accepted
range of values for the ratio [44–47].
3.3. Jump into self-contact (looping)
Under increasing D the R = 2 solution emerging from the 5rst-order buckling mode in Fig. 3
becomes planar at the ‘into-the-plane’ critical point indicated by the diamond and given by a critical
modulus kin in (32). At that point the solution has lost exactly one turn of twist and is twistless.
The dependence of this critical point on the torsional sti8ness of the rod is illustrated by the dashed
curve in Fig. 4 which was obtained by Zajac [1] in parametrised form as
D
C
K(k)[(4 − 3k 2 )K(k) + (−4 + k 2 )E(k)]
E(k)
2
;
=
(37)
=2− 2 1−
L
k
K(k)
B
(1 − k 2 )K 2 (k) + (2 − k 2 )K(k)E(k) − 3E 2 (k)
177
from analysis of the 3D variational equations about the clamped planar looped elastica. The corresponding critical load (for an arbitrary n-mode!) is found by referring to (32). As C=B tends to
in5nity D=L tends to 0 (and T to in5nity), i.e., the planar elastica retains the loop for all end loads.
As Zajac observes, the insensitivity of this pop-out point for the twistless planar elastica to C=B
means that one cannot greatly mitigate the formation of kinks by designing cables with favourable
C=B ratios. But this assumes the cable has zero thickness and no twist; we shall consider pop-out in
twisted rods of 5nite thickness in Section 5.1. This Zajac pop-out of a zero-radius rod, associated
with a loss of stability of the untwisted planar elastica under controlled decreasing D, can be seen
as a supercritical pitchfork bifurcation.
The R = 2 solution curve separates di8erent post-buckling behaviours. Curves emerging from
5rst-order modes with R ¿ 2 form limit points in D, where, under rigid loading, the rod dynamically
jumps into a self-contacting looped solution (to be considered in Section 5). If we could arti5cially
follow through the entire (stable and unstable) post-buckling path for these solutions, to zero D and
in5nite T , we would remove exactly one turn worth of twist.
Looping of rods with residual twist, i.e., those with more than one turn of end rotation put in,
was studied by Coyne [3]. He took a ‘semi-in5nite’ approach in which the rod is taken to be of
5nite length but of the shape of the homoclinic solution of the equations (i.e., the solution which
is asymptotically straight towards both ends and strictly needs in5nite length). Since an in5nite rod
always e8ectively experiences dead load he found for the looping instability the in5nite-rod Greenhill
[37] condition m2z = 4t. See also [48] for a more systematic study of the semi-in5nite approximation
and how it relates to the 5nite-length case.
Besides the out-of-plane bifurcation (for R = 0) and the into-the-plane bifurcation (for R = 2)
there is a third bifurcation associated with the planar elastica, again for R = 0. It occurs along the
second-mode branch of the in6ectional planar elastica (see Fig. 2(a)), and de5nes the point where
the R = 0 curve connects to that planar elastica curve. (So the spatial elastica curve for R = 0 is seen
to connect the 5rst-mode in6ectional planar elastica to the second-mode in6ectional planar elastica,
both of which are independent of .) For the present value of this second-mode bifurcation occurs
too close to D=L = 2 to be visible along the (a) curves in Fig. 3. It can be observed in Fig. 6 for
= 1:3 and 1.7. This value 1.7 is larger than c (for n = m = 1) and generates di8erent behaviour
of solution curves through the ring. For instance, additional limit points develop along these curves,
leading, for R = 0:3, to two stable branches of solutions.
3.4. Snap-through
In Section 3.2 we saw a pitchfork-type secondary bifurcation where the rod buckles out of the
plane. Put into energy terms: it is favourable for the rod to release some bending energy at the
cost of some torsional energy. It is well-known from bifurcation theory that if a second parameter
in the system is varied, a pitchfork bifurcation gets unfolded (generically) into solution curves with
a simple fold (see the curves labelled (b) in Fig. 3). In structural engineering this is known from
systems in which imperfection parameters are present. In our rod the end rotation plays this role as
secondary bifurcation parameter, and the fold gives rise to snap-through behaviour in which the rod
dynamically jumps from one con5guration to another, symmetrically related, con5guration. Associated with this are possibilities of hysteresis loops in which the rod jumps back and forth between two
paths.
178
2
2
γ = 1.7
γ = 1.3
1
TL2/4π2B
TL2/4π2B
1
0
(s)
(r)
-1
(s)
(r)
(q)
0
(t)
(s)
(r)
(q)
(p)
-1
(q)
(p)
-2
-2
-3
-3
0
(a)
0.5
1
D/L
1.5
0
2
0.5
(b)
1
D/L
1.5
2
Fig. 6. D–T diagrams showing the second-mode bifurcation: (a) = 1:3 and curves are for R = 0 (p), 0.628 (q), 1.885 (r)
and 3 (s); and (b) = 1:7 and curves are for R = 0 (p), 0.1 (q), 0.3 (r), 0.6 (s) and 1 (t). Solid branches are stable, dashed
branches unstable. The squares indicate the second-mode bifurcation, the triangles the out-of-plane bifurcation, while the
circles mark the ring.
1.5
1
ML/2πB
0.5
d=0.5
0
0.62
-0.5
0.8
-1
1.2
0.95
-1.5
-10
-5
0
R
5
10
Fig. 7. Load–de6ection diagram with solution curves for D=L = 0:5, 0.62 0.8, 0.95 and 1.2. These curves have to be
imagined closed by identifying the end points at R = ±2, where the rod is a planar self-intersecting non-in6ectional
elastica. At the origin the rod is an in6ectional elastica. Solid lines represent stable branches, dashed lines unstable
branches. ( = 5=7).
The phenomenon is illustrated in the M -R load-de6ection diagram of Fig. 7. All curves are running
from (R; M ) = (−2; 0) to (R; M ) = (2; 0). These end points have to be identi5ed and the curves
imagined closed: at (R; M ) = (2; 0) the rod is a planar non-in6ectional elastica with no twist and
with a self-intersection (for D=L ¿ 1 the hypothetical closing piece connecting the ends of the rod
intersects itself), and when going through this self-intersection R jumps from 2 to −2 (and the
writhe of the hypothetical closed centreline from 1 to −1). At the origin the rod is a 5rst-mode
in6ectional planar elastica (for instance, a 5gure-of-eight for D=L = 1).
179
Fig. 8. Scaling behaviour under a change of of solution curves in the R–m diagram, where m = ML=(2B). The straight
lines are given by m = R=(2). The two double arrows have equal lenghts.
Snap-through is possible for dout =0:5590 ¡ D=L ¡ 0:936 and D=L ¿ 1:109 when there is a branch
of stable solutions available as one jumps o8 the fold. The limits 0.936 and 1.109 are determined
numerically and correspond to values of D=L for which the fold occurs precisely at R = 2. For
0:936 ¡ D=L ¡ 1:109 the rod will jump into a self-contacting solution at some |R| ¿ 2. However,
in Section 5 we shall see that even within the range of parameters where snap-through is possible
in principle, jumps to self-contact may occur because of self-contacting solution branches interfering
with the curves in Fig. 7.
Note that for values of D=L smaller than that of the into-the-plane instability (D=L = 0:6767), the
curves in Fig. 7 also show the fold instability with jump into self-contact discussed in the previous
section.
As in Fig. 3, Fig. 7 shows only curves with stable branches. Other curves exist at the same value
of D=L. For example, look ahead to the dashed curve in Fig. 10(c) drawn there because a (stable)
branch of self-contacting solutions bifurcates from it. This dashed curve, for D=L = 1:2, runs from
(R; M ) = (−4; 0) to (R; M ) = (4; 0), at which points the solution is a second-mode non-in6ectional
(i.e., 2-looped) elastica.
3.5. A scaling law for R and In Fig. 5 we varied to study what happens when the ring goes unstable. In fact, as pointed out
in Reference [12], there exists a scaling law which allows one to obtain the behaviour for any from knowledge about solution curves for any particular . The rule, explained in Fig. 8, is based on
the observation that the di8erence W := R − ML=C = R − 2m=, m := ML=(2B), like the writhe Wr
to which it is related, is only a function of the centreline of the rod and therefore independent of .
Thus, if for = 1 the solution curve in the R–m plane is known, then the solution curve for = 2
can be found by translating R, at 5xed m, by an amount equal to the horizontal distance between
the straight lines of slope 1 =(2) and 2 =(2). It may be convenient in calculations to take for the value 1, so that the equations for d3 and d1 , (9) and (11), become identical, with just di8erent
initial conditions.
180
In view of this scaling law, the complicated spiralling behaviour of Fig. 5 for 5xed R and varying
must have an analogue in the case that is kept 5xed and R is varied.
4. Formulation of a rod with point contacts
The thickness of the rod, and the fact that a real rod cannot intersect itself, can be taken into
account on the level of 1D rod theory by considering an impenetrable elastic tube of certain radius
about the centreline of the rod. This technique was applied by Shi and Hearst [23] and Miyazaki
and Kondo [5] to con5gurations with a single point of contact, and by Coleman and his co-workers
(see [11,12,49] and references therein) to more general situations. Most of these studies, but not
Reference [5], are concerned with supercoiling of DNA plasmids, i.e., closed pieces of DNA. In
one study in Reference [49] open rods are considered, but subject to pinned boundary conditions
as de5ned in Section 3.1, re6ecting the situation in recent experiments on segments of individual
DNA molecules placed in a magnetic trap [50]. (The author actually calls it Cardan-joint boundary
conditions but these are more commonly taken to mean that no moment can be transmitted about
one 5xed axis and one body axis; see, e.g., Section VIII.12 of [17].) Both pinned and Cardan-joint
boundary conditions are rarely encountered in engineering applications. In Reference [5] a clamped
strut is considered.
Here we apply the same technique to our clamped rod. We only consider con5gurations with
isolated points of contact. Coleman and Swigon [12] also studied rods with intervals of continuous
self-contact for the special case of a straight line of contacts. Such con5gurations are encountered in
ply structures seen in DNA molecules and twisted textile yarns, and in Reference [12] were found
to be the natural state reached after going through a number (two or three) of point contacts. We
have something to say about rods contacting along an interval in a separate publication [15].
We follow the analysis of Section 2 and choose the origin of arclength in the middle of the
rod. Although in case of one or several points of self-contact there may be non-reversible solutions
satisfying clamped boundary conditions (there were for pinned boundary conditions in Reference
[49]), we shall again only look at reversible solutions having m0y = 0. At a point of self-contact
(assumed to be frictionless) a reactive concentrated force will act between two non-neighbouring
cross-sections of the rod. As usual, this force will be assumed to be normal to the surface of the
rod. Consequently, the constant internal force F will undergo a discontinuous change. The moment
M , on the other hand, will be assumed to be continuous (i.e., no concentrated moments), as will
be R and the directors di (and hence u3 ). From Eq. (7) we see that in order to ensure continuity
of M across a point of contact, M0 will have to change. Thus we consider equations (3), (9) and
(11) with subscripts and write
R = d3 ;
(38)
Bd3 = (Fi × R + Mi0 ) × d3 ;
(39)
Bd1 = (Fi × R + Mi0 ) × d1 + (B − C)u3 d3 × d1 :
(40)
Here Fi and Mi0 are the as yet unknown constant internal force and matching moment in the section
[Si ; Si+1 ], where S0 = 0, Sn+1 = L=2, and n is the number of contacts. (The property of having normal
181
reactive forces and no concentrated moments has been rigorously established for a wide class of
contact problems by Schuricht [51].)
We shall only consider solutions with symmetric contacts, i.e., contacts at S-symmetric points
R(S) and R(−S). These include the con5gurations the rod jumps into at the limit points in Fig. 3,
as well as the writhing con5gurations obtained when subsequently moving the ends in. At the 5rst
point of contact, S1 , we have the geometrical relations
|R(S1 ) − R(−S1 )| = 2L;
(41)
(R(S1 ) − R(−S1 )) · d3 (±S1 ) = 0;
(42)
where is the dimensionless radius of the rod. Since F is de5ned to be the force from the element
at S acting on the element at S, where S ¿ S, we have
F0 = F1 + NF1 ;
(43)
with
NF1
(44)
(R(S1 ) − R(−S1 ))
2L
the force, of magnitude NF1 , pointing from S = −S1 to S = S1 . For M to be continuous we need
NF1 =
M (S1 ) = F0 × R(S1 ) + M00 = F1 × R(S1 ) + M10 ;
(45)
which yields
M10 = M00 + (F0 − F1 ) × R(S1 ) = M00 + NF1 × R(S1 ):
So we can write
(46)
NF1
(X (S1 ); 0; Z(S1 ));
(47)
L
NF1
M10 = (M00 x ; 0; Mz ) +
(48)
(−Y (S1 )Z(S1 ); 0; X (S1 )Y (S1 )):
L
The same can be done at each contact and we obtain
NFi
(X (Si ); 0; Z(Si ));
(49)
Fi = Fi−1 −
L
NFi
(−Y (Si )Z(Si ); 0; X (Si )Y (Si ))
Mi0 = M(i−1)0 +
(50)
L
as the constants to be used in (39), (40).
Thus in our shooting method the jump conditions (49) and (50) give two extra equations for
the two unknowns NFi and Si for each contact-free section of rod. Together with the two clamp
conditions (17) applied to the terminal section, and the equations (23) and (one of) (24) for the
end shortening and end rotation this gives a well-posed problem of 2n + 4 equations with the same
number of unknowns for a (symmetric) solution with n points of self-contact. This count may be
compared with the 12n + 6 according to the method used in Reference [12] for solving full rather
than half solutions.
Since at contact points M is continuous while F has a jump in a direction orthogonal to d3 , we
see that of the 5rst integrals identi5ed in Section 2.1 the twist u3 and the Hamiltonian M ·u=2+F ·d3
will be conserved along the entire self-contacting rod, while f and mz will jump at contact points.
F1 = (0; 0; F) −
182
5. Numerical results for self-contact
40
30
30
25
(∆ F)L2/4π2B
TL2/4π2B
As before, we take = 5=7, and we further 5x = 0:03=(2). Results for 5xed end rotation are
shown in Fig. 9. It is found that for R less than about 4:8 the rod delocalises under increasing
end shortening D with the 1-contact solution approaching the ring. For R larger than about 4:8
curves in the D–T diagram form limit points in D. Physically, the loop starts to rotate and the rod
writhes up into a ply. Note that the 1-contact solution goes into compression before it becomes
unstable and jumps into a state with more contacts. Diamonds along the unstable branches indicate
where self-penetration starts to occur. No bifurcating branch of solutions could be detected, but the
numerical approach of this self-penetration suggests that a branch of solutions with an interval of
self-contacts might bifurcate, which we cannot pick up with our present techniques. This is di8erent
from the constant-D case discussed below where a branch of 2-contact solutions bifurcates o8 the
1-contact branch. Fig. 9(b) shows how the force NF varies with D.
Results for 5xed end shortening are shown in Figs. 10–12. In Fig. 10(a), for D=L = 0:5, we 5nd
solutions with up to 3 points of self-contact. The transition from the contact-free to the 1-contact,
and the transition from the 1-contact to the 2-contact solution, are through dynamic jumps at folds,
while the transition from 2-contact to 3-contact is smooth. True-view 3D shapes of the rod at these
events are shown in Fig. 12. Extensive numerical searches did not give any hint of the existence
of a branch of 4-contact solutions as R is increased further. This is consistent with the work of
Coleman and Swigon [12] who 5nd, for closed rods, that at some point the third self-contact grows
into an interval of self-contacts, still 6anked by two points of self-contact.
Fig. 10(b) shows that for D=L=0:86 multistability occurs: at the fold along the (solid) contact-free
curve the rod can either snap-through and remain without self-contacts, or jump into a con5guration
with one point of self-contact. (Of course, we cannot rule out the existence of further stable solutions.) Experiments reveal that the contact-free option is normally taken. In fact, the 1-contact curve
20
10
0
15
10
5
-10
-20
0
0
(a)
20
0.2
0.4
0.6
D/L
0.8
1
0
(b)
0.2
0.4
0.6
0.8
1
D/L
Fig. 9. Writhing a rod or not, by varying D=L and 5xing R at 4; 5; 6 and 8. For R = 4 the rod does not writhe
and the curve of 1-point contacts runs up to the ring at D=L = 1 (indicated by the circle). In contrast, the curves for
R = 5; 6 and 8 form a limit point, signifying that the rod writhes up into a ply. The transition to writhing takes place
between R = 4:8 and R = 5. Triangles indicate pop-out points where NF drops to zero (see (b)) and self-contact is
lost. Diamonds indicate points where, presumably, an interval of self-contact is formed. ( = 5=7; = 0:03=(2).)
1.5
1.4
(d)
1.2
1
1
(b)
0.8
(f)
(e)
0.6
0.5
ML/2πB
ML/2πB
183
(c)
0.4
0
-0.5
0.2
0
-1
(a)
-1.5
-10
-0.2
0
(a)
5
10
15
R
20
25
30
-5
0
5
R
(b)
10
15
20
2
1.5
ML/2πB
1
0.5
y
0
y
y
z
z
z
z
-0.5
x
-1
-1.5
-2
-15
(c)
-10
-5
0
R
5
10
15
Fig. 10. Ply formation under varying R and (a) D=L = 0:5; (b) D=L = 0:86; and (c) D=L = 1:2. All the lower branches
coming in at the various folds are unstable. Triangles indicate pop-out points where one self-contact is lost/gained. In (a)
curves for up to three points of self-contact are included. Labels refer to the 3D shapes shown in Fig. 12, while the inset
shows that the 2-contact curve forms a limit point to the left. In (b) the dashed curve is for 1-contact solutions. Insets
give true z–y views of the rod’s centreline at the indicated points. In (c) the 1-contact curve bifurcates from the dashed
curve of non-contacting solutions running from (R; M ) = (−4; 0) to (R; M ) = (4; 0), at which end points the solution is
an (inverted) second-mode non-in6ectional elastica. ( = 5=7, = 0:03=(2)).
goes round a fold and intersects the free rod curve a second time at R = −1:2841 where again NF
drops to zero. In view of the fold rule this ‘pop-out’ only involves unstable branches. Upon increase
of D=L the two pop-out points coalesce at D=L = 0:8946, leaving a contact curve disconnected from
the contact-free curve.
Multistability is also seen in Fig. 10(c), for D=L = 1:2 (i.e., for inverted solutions). Experiments
show that this time the contact-free route is normally taken. The branch of 1-contact solutions does
not bifurcate from the usual contact-free curve but from an entirely unstable contact-free curve
(dashed in the 5gure) which at (R; M ) = (±4; 0) connects to a second-mode non-in6ectional planar
elastica. The solution at R=0 along this dashed curve is, geometrically, close to a 5gure-of-eight held
at the edge (for D=L = 1 it would be an exact 5gure-of-eight). The dashed curve has an additional
intermediate zero of M where the solution is non-planar.
184
0.0108
0.4
0.2
0.0106
0.3
0.0104
0.1
0.01
x
0.2
y
∆
0.0102
-0.1
0.0096
0
0.0094
-0.2
0.0092
0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23
(a)
s
-0.1
-0.2
-0.1
0
0.1
0.2
0.1
0.2
-0.2
-0.1
0
0.1
0.2
0.1
0.2
0.2
0.1
0.0102
0.01
x
0.2
y
∆
0
z
0.3
0.0104
0
0.1
0.0098
-0.1
0.0096
0
0.0094
-0.2
0.0092
0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23
-0.1
-0.2
-0.1
s
0
0.1
0.2
z
z
0.0108
0.4
0.2
0.0106
0.3
0.0104
0.1
0.0102
0.01
x
0.2
y
∆
-0.1
0.4
0.0106
0
0.1
0.0098
-0.1
0.0096
0
0.0094
-0.2
0.0092
0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23
(c)
-0.2
z
0.0108
(b)
0
0.1
0.0098
s
-0.1
-0.2
-0.1
0
z
0.1
0.2
-0.2
-0.1
0
z
√
Fig. 11. Formation of the third point of contact along the 2-contact branch in Fig. 10(a). - = 2 x2 + z 2 is the symmetric
self-distance of the rod; s is arclength. The minimum possible value of - is 2, attained at self-contact, and is indicated
by the dotted horizontal line in the left 5gures. The right-most self-contact turns out to have the largest pressure force
NF. The projections of the shape of the rod (centre and right) are true views (from in5nity) and show the rotation of
the end loop. ( = 5=7; = 0:03=(2)).
Fig. 11 illustrates the formation of the third point of contact along the 2-contact branch in Fig.
10(a) using the symmetric self-distance
(51)
-(s) = [x(s) − x(−s)]2 + [y(s) − y(−s)]2 + [z(s) − z(−s)]2 = 2 x(s)2 + z(s)2
to monitor the process. Notice the rotation of the end loop relative to the rig as R is increased.
(a)
(b)
(c)
(d)
(e)
(f)
185
Fig. 12. True 3D views of the rod along stable solution paths in Fig. 10(a). Labels correspond to the labels in that 5gure.
The starting point (a) is at the planar elastica. The 5rst jump, into 1-point contact (from (b) to (c)), occurs at R=11:6315;
the second jump, into 2-point contact (from (d) to (e)), occurs at R = 18:4027. The transition to 3-point contact at (f)
(corresponding to (c) in Fig. 11) is smooth.
5.1. Pop-out
Yabuta et al. [2,52] studied the stability of thin looped self-contacting rods by looking at the
second variation of the total potential energy. By assuming one turn of a circular helix as the shape
of the loop, and a perfectly straight rod elsewhere, a stability criterion could be derived. A simpli5ed
criterion was obtained by further assuming that all the twist in the rod is removed by looping (i.e.,
no residual twist) and taking zero rod thickness, but this gives non-physical results in sharp contrast
with the exact result (Eq. (37) above) by Zajac (such as the prediction of a critical above which
no pop-out occurs in the planar elastica, which therefore would kink, or ‘hockle’).
186
‘exact’ numerics
homoclinic approx.
Yabuta
50
2
TL2/4π2B
TL2/4π2B
40
30
20
1
0.5
0
0.4 0.5 0.6 0.7 0.8 0.9
D/L
10
0
0.05
1.5
0.1
0.15
0.2
0.25
D/L
0.3
0.35
0.4
Fig. 13. Pop-out locus parametrised by R. Markers are for R = 2:5, 3, 3:5, 4, 5, 6, 7 and 8 (in increasing order
of tension). The solid curve is obtained from our criterion - = 2, the long dashed curve represents results based on the
homoclinic orbit, while the short dashed curve shows Yabuta’s approximation. The inset shows further results with markers
for R = 2:25 and R = 2:1. For R = 2:25 and higher the homoclinic results are very accurate. ( = 5=7; = 0:03=(2)).
If the looped rod has residual twist, i.e., if R ¿ 2, then at the point of self-contact a force will
act to maintain the contact. If the end shortening is decreased by pulling the ends (keeping R 5xed)
the loop will be tightened up to a point where the loop pops open. This pop-out only occurs in rods
of 5nite thickness; an in5nitely thin rod with R ¿ 2 would not go unstable and could be pulled out
straight. In many practical cases a cable or pipe will kink (i.e., su8er plastic deformation) before
pop-out is reached, damaging the structure.
Zajac [1] studied pop-out of a clamped rod of zero thickness by analysing the 3D variational
equations about the clamped planar looped elastica. The result is the into-the-plane critical curve
we have already seen in Fig. 4. Indeed, Coyne [3] points out that the Zajac condition (37) is best
interpreted as a criterion for looping rather than pop-out. Our analysis shows precisely how Zajac’s
result (for zero-radius rods) relates to both, since it applies to R = 2 only, at which point looping
and pop-out become the same phenomenon evolving in opposite directions.
Pop-out may also occur in rods with multiple points of self-contact when one contact point is lost
(as in Fig. 10(a)). At the point of pop-out (out of a single contact) the shape of the rod is again
governed by the contact-free rod equations of Section 2.1. Therefore, in the numerical continuations
which produced Fig. 9 we can infer pop-out when the self-distance - reaches 2. The locus of
pop-out points thus obtained is shown by the solid curve in Fig. 13 on which some R-values are
highlighted. Along a 1-point contact curve in Fig. 9 pop-out is de5ned by the intersection point of
this curve with the free rod curve at the same value of R. At this point the force jump NF drops
to zero (see Fig. 9(b)). For a rod of zero radius pop-out is described mathematically by a pitchfork
bifurcation; at 5nite radius it becomes a type of discontinuous bifurcation.
A semi-analytical estimate of the conditions at pop-out can be obtained by following (and slightly
improving upon) Coyne [3] in using the homoclinic solution of the rod equations already mentioned
in Section 3.3. For this homoclinic solution the shape of the rod is given by (see, e.g., Reference
[23]):
1 x(s) =
4t − m2z sech(s 4t − m2z ) sin(mz s);
2t
1 4t − m2z sech(s 4t − m2z ) cos(mz s);
2t
1 4t − m2z tanh(s 4t − m2z );
z(s) = s −
2t
with the self-contact occurring at arclength s = sV such that
187
y(s) = −
z(s)
V = 0;
x(s)
V = :
Further, in Reference [22] the expressions
M
ML
2
+ 4 arccos √
and R =
D = (2=T ) 4BT − M
C
2 BT
(52)
(53)
(54)
are derived for the end shortening and end rotation of the homoclinic solution. From (52)–(54) we
can 5nd the tension and end shortening at pop-out as a function of the radius of the rod, , and the
amount of end rotation, R, applied. The
√ result is the long dashed curve in Fig. 13. Also displayed in
this 5gure are results (having d = 1= 2t) following from Yabuta’s analysis taking u3 = (R − 2)=L
for the residual twist. (For the Yabuta curve, Eq. (34) in Reference [52] is used, which gives better
agreement than the corresponding but con6icting result quoted in Reference [2].) It is seen that for
R ¿ 2:25 the results based on the homoclinic approximation are virtually indistinguishable from the
‘exact’ numerical results. Notice that the pop-out curve in the D–T diagram (in fact any of the three
curves in Fig. 13) considered as a whole is independent of ; its parametrisation by R, of course,
does depend on .
Depending on whether the 1-contact branch emanates from an unstable or stable branch of
contact-free solutions, at pop-out the rod undergoes either a dynamic jump or a smooth transition out of contact. The former is the case in Fig. 9(a) where the rod jumps to a contact-free state
at much lower tension (or perhaps compression) T described by the point vertically down from the
pop-out point on the same constant-R curve. Note from Fig. 9(b) that NF goes through a maximum
before it rapidly drops to zero and pop-out occurs. An alternative way of jumping out of contact is
provided by the small-scale limit point seen along the 2-contact branch in Fig. 10(a): if one were
to decrease R along this branch the rod would jump out of 2-point contact before the force NF
dropped to zero.
An example of smooth pop-out is seen in Fig. 10(b).
5.2. Rig-wrench behaviour
As we have seen in our formulation, the forces and moments acting across any cut through a
5nite-length rod will be equivalent to a wrench acting about a ‘wrench axis’ in space. It is for this
reason that any rod of 5nite length, deforming in three dimensions under end forces and moments,
will lie conceptually within the 5eld of solutions (the spatial elastica) that can be adopted by
an in5nite rod loaded by equal and opposite wrenches. This makes the spatial elastica a natural
framework within which to explore and understand the behaviour of 5nite rods. We do, however,
have to take cognisance of the fact that during loading the wrench of the elastica solution, which
before buckling lies along the axis of the testing rig, will move spatially relative to this ‘rig axis’
during post-buckling.
188
(a)
(b)
Fig. 14. Examples of a contact-free (a) and a 2-contact and (b) rod con5guration with the equivalent wrench lines included.
Also shown are the projections of these lines on the plane parallel to the x–z plane and going through both ends of the
rod. The wrench line of a (piece of) free rod is the line about which the internal force and the equivalent moment act.
The straight lines in Fig. 14 indicate the equivalent wrench lines for a contact-free rod as well
as for (contact-free pieces of) a rod with two points of self-contact. The wrench line undergoes a
step change at a contact point on account of the step change in the internal force f. The sequence
of plots in Fig. 11 also nicely illustrate the movement of the wrench relative to the rig during the
loading process (recall that ez was chosen along the wrench axis). This movement is composed of
a translation h in the y direction and a rotation & in the x–z plane given by
M0x
(55)
h=
; cos & = d3z (L=2):
F
This may be compared with the situation for the clamped planar elastica which, as shown by the
insets in Fig. 2(b), has a pure translation for the odd modes and a pure rotation for the even modes.
For a well-localised solution in a long rod the wrench line is close to the line of the rig.
6. Experiments
In this section we report on some laboratory experiments we carried out to supplement our theoretical investigations of the various jump phenomena. Surprisingly few experimental studies exist
in the literature. We can only mention Yabuta’s study [52] of pop-out for which jacketed optical
5bres were used because of their high 6exibility. However, the author mentions neither the exact
boundary conditions nor the applied loading sequence.
The choice of material for our experiments was constrained by the requirements that the rod must
be highly 6exible in bending and twisting but must not su8er from sag due to self-weight. Also, for
a quantitative study the loads involved must be measurable (which e8ectively excludes rubber rods).
The superelastic nickel titanium alloy nitinol proved to be a suitable material. Nitinol has found
extensive use in a wide range of applications. For instance, in the medical industry it is used in
orthopaedic devices and orthodontic arches. It is also employed as spectacle frames. The properties
of nitinol are strongly dependent on processing history and a wide range of values for B are given by
di8erent manufacturers. Therefore, the elastic rigidities, B and C, were estimated directly by means
of simple cantilever and torque tests. These gave linear constitutive relations and a ratio of = 5=7.
Experiments were limited to the range 0 6 R 6 5. Within these limits the specimens remained in
the linear elastic range.
Circular-cross-section nitinol rods with a radius of 0:5 mm and ranging in length from 350 to
500 mm (from 1000 to 1200 mm for inverted experiments) were clamped in a specially constructed
189
1
3
0.8
2
0.6
0.4
0.2
ML/2πB
TL2/4π2B
1
0
0
-0.2
-1
-0.4
-0.6
-2
-0.8
-3
0
0.2
0.4
0.6
1.2
1.4
1.6
1.8
-1
2
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-0.4
-0.6
- 0.6
-0.8
- 0.8
-1
R rads
5
10
5
10
5
10
-0.2
-0.4
0
0
0
-0.2
-5
-5
R rads
1
-10
-10
(b)
D/L
1
(c)
1
ML/2πB
ML/2πB
(a)
0.8
(d)
-10
-5
0
R rads
Fig. 15. Experimental results for a clamped nitinol rod superimposed on theoretical curves: (a) tests at 5xed R for R = 0,
, 2, 3, 4 and 5 (including tests on inverted rods) with curves of Fig. 3; (b,c,d) tests at 5xed D for D=L = 0:5,
0.677 and 0.8, respectively, with curves of Fig. 7. Notice the hysteresis loops (indicated by the arrows) associated with
both snap-through and pop-out. ( = 1=900).
rig (see Fig. 1) in which their ends could be rotated and displaced independently. By means of
suitably calibrated transducers 5xed in line at the ends of the rod the rig moment M and rig force
T could be measured to an accuracy of 0:0001 Nm and 0:0001 N, with values ranging from −0:2
to 0:2 Nm and −0:5 to 1 N, respectively. Some experiments involved the recording of loads on the
limit of the sensitivity of the transducers. However, with care in selecting lengths of rod (short
enough to avoid important sagging e8ects, yet long enough to have an appreciable linear elastic
range) good repeatability of results was obtained. Each of the data points shown in Figs. 15(a) and
(b) represents an average over at least three and at most 5ve measurements.
Starting with a straight untwisted rod, two sets of experiments were performed. In one set the end
rotation, R, was input and kept 5xed while the end shortening, D, was controlled, i.e., the aligned
190
ends were moved towards each other in small increments. For each adjustment of D the end tension,
T , and twisting moment, M , were recorded. Some results are shown in Fig. 15 where it can be seen
that reasonably good correlation between the mathematical model and the experiment is obtained.
For R=0 and 2 the pitchfork bifurcations to and from a planar state occur as predicted theoretically.
For R ¿ 2 the values of D at which the rod jumps into a loop are also in good agreement with
the theory. After loop formation experiments were continued under both increasing and decreasing
D. It is found that by increasing D the loop remains intact up to values of R ≈ 4:5 with the shape
of the loop evolving into a more circular form as D approaches L. However, for R ¿ 4:5 the loop
itself rotates to form a ply, because the loop does not release enough torsional strain energy during
loop formation. Theoretically the transition to writhing was located near 4:8 for a slightly di8erent
slenderness ratio (see Section 5).
It is also possible to induce the rod to pop out of the loop by decreasing D. In such an experiment
the loop dimensions diminish causing the curvature, and therefore the bending moment around the
loop, to increase. This can cause permanent plastic damage (i.e., kinking). In fact, it was found that
a completely successful (i.e., kink-free) pop-out experiment is only possible for a small amount of
end rotation (up to about 0.1 radian above R = 2).
In the second set of experiments the initially straight rod was 5rst bent by an input of D, and
the experiment was continued under control of R keeping D 5xed. For D less than that at the
out-of-plane bifurcation (D=L = 0:5990) the only jump encountered is into a loop at high R. This
jump corresponds to the fold for R ¿ 2 in the previous set of experiments. However, once D
reaches the critical value at which the out-of-plane bifurcation occurs, con5gurations in the vicinity
of R = 0 become unstable and snap-through occurs. The rod jumps through the unstable region, and
if R is reversed it does not jump back again at the same value, i.e., an hysteresis loop emerges. As
the end shortening is increased, the extent of this hysteresis cycle spreads, and with it the magnitude
of the snap buckling. However, this development is accompanied by a corresponding diminution of
the jump into (and out of) a self-contacted loop (which all takes place at much higher values of R
than snap buckling).
At D=L = 0:6767 the jump into a loop 5nally disappears and is replaced by a smooth path both in
and out of self-contact. Some results are shown in Fig. 15(b) – (d) where the changing size of these
jumps for four di8erent settings of D=L can be clearly seen. Also in Fig. 15 the hysteresis cycles
associated with the snap-through and the jumping in and out of a loop are clearly illustrated.
In the vicinity of D=L=1 the rod behaves like a ring. A twisted ring becomes unstable at a critical
twisting moment given by the Zajac condition (35) and jumps into a 5gure-of-eight with two points
of self-contact. Reversing the experiment causes the rod to jump to a stable 5gure-of-eight with one
point of self-contact, creating a small hysteresis cycle, before jumping out of contact back into a
ring. This behaviour is in agreement with the theoretical results in Reference [12] (although these
are for a slightly di8erent value of ). Unfortunately, the rig does not allow load measurements in
this region. As D is increased beyond it the loads are measurable again and under control of R the
rod jumps from one side of R = 0 to the other, but without self-contact.
In conclusion, most of the jump phenomena and associated hysteresis loops found theoretically
can be observed, with reasonably good quantitative agreement, in relatively simple laboratory tests.
Deviations from the theory are likely to be the result of the e8ect of gravity and sensitivity to small
errors in satisfying exact clamped boundary conditions. For instance, experiments with anisotropic
rods (i.e., ones for which B1 = B2 ) suggest that the vertical o8set between the theoretical curve and
191
the experimental data at primary buckling (on the vertical axis in Fig. 15(a)) might be caused by
gravity. A strongly anisotropic rod will tend to bend only about its weakest principal axis, either up
(hogging) or down (sagging), and experiments reveal a di8erence in behaviour depending on which
direction the rod buckles in. Details of these and other experiments can be found in Reference [53].
7. Discussion
We have made a comprehensive study of jump phenomena in clamped rods with and without
self-contact. A summary of the results is given in Fig. 16 where D–R displacement diagrams are
shown for three typical values of . The curves are loci of fold and pop-out points obtained by
numerical continuation (pop-out being de5ned by the condition - = 2). Bifurcation points are
indicated by the same markers as used in the main text. Diagram (a) is for = 5=7, the value for
nitinol used throughout. Diagram (b) is for = 1:7 just larger than c in (36) (for n = m = 1). Notice
that the fold lines near D=L = 1 have formed some cusps leading to more complicated behaviour.
12
12
fold
8
0.6
0.4
0.2
Rc
0
6
pop-out
pop-out
4
2
6
R
R
0.8
Rc
8
0.9 0.95 1
pop-out
2
γ = 5/7
1.05 1.1 1.15 1.2
D/L
pop-out
4
fold
0
γ= 1.7
fold
0
0
(a)
1
10
fold
R
10
0.2
0.4
0.6
0.8
1
1.2
1.4
0
1.6
D/L
0.2
0.4
0.6
(b)
0.8
1
D/L
1.2
1.4
1.6
12
10
R
R
fold
8
6
1.5
1
0.5
0
Rc
-0.5
-1
-1.5
0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
D/L
pop-out
4
pop-out
2
γ=2.1
fold
0
0
(c)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
D/L
Fig. 16. Diagrams summarising the various jump phenomena for: (a) = 5=7; (b) = 1:7; and (c) = 2:1. Triangles mark
the out-of-plane bifurcation, diamonds the into-the-plane bifurcation and squares the second-mode bifurcation.
192
Diagram (c) is for = 2:1. For this value of the out-of-plane bifurcation occurs for D=L ¿ 1 (cf.
Fig. 4), i.e., for inverted solutions, and the out-of-plane and second-mode bifurcations have changed
order. The diagrams can be used to locate regions in the displacement plane where stable solutions
exist.
We have only looked at clamped boundary conditions. Although they occur widely in engineering
applications they are by no means the only ones found, and it is clear that the bifurcation behaviour
observed depends strongly on the type of boundary conditions considered. For instance, for clamped
rods we found the circular ring to be an organising centre of bifurcation behaviour; yet for instance
for ball-and-socket boundary conditions (i.e., M (L=2) = 0) the ring cannot even be supported. Meanwhile, the solutions for a rod whose ends are held in a sleeve form a subset of the solutions for a
clamped rod, namely those that have no twist, i.e., u3 = 0 (cf. case 8 in Table 2).
Our analysis does not depend on the length scale L of the problem and therefore applies to
DNA molecules, carbon nanotubes [54] and submarine communication cables [3] alike. However,
gravity e8ects, ignored in this study, do depend on the length scale. Consider for example the
relative de6ection (de6ection/length) due to self-weight of a horizontal rod of circular cross-section
of radius r. Engineering bending theory shows that this scales as (g=E)(L=r)2 L, where and E are
the density and Young’s modulus of the material, g is the gravitational acceleration and L is the
length of the rod. To minimise gravitational e8ects in an experiment we therefore want a material
with low =E and a low aspect ratio, L=r, consistent with still having a ‘thin’ rod. More signi5cantly,
we want an absolute low value for L. Relative gravitational de6ections are small for physically small
objects, large for physically large objects. This is why we cannot have land animals much larger
than elephants or much taller than gira8es.
Acknowledgements
G.H. was supported by a Royal Society Research Fellowship, S.N. by a Royal Society Visiting
Fellowship.
Appendix A. Clamped solutions are reversible
We show that clamped solutions to the equilibrium equations (3), (9) are reversible. This generalises a result by Domokos [55] stating the same for closed rods. Introducing Euler angles and
to parametrise the unit vector d3 as
d3x = sin cos ;
d3y = sin sin ;
d3z = cos ;
(56)
and using the conserved quantities discussed in Section 2.1, we can reduce (3), (9) to the following
system of (dimensionless) equations:
+
=
dV ()
= 0;
d
mz − m3 cos ;
sin2 V () =
(mz − m3 cos )2
+ f cos ;
2 sin2 (57)
(58)
2
2
1.5
1.5
O
1
1
E
D
T
0
0.5
S
θ′
θ′
0.5
B
C
A
-0.5
0
-0.5
-1
-1
-1.5
-1.5
-2
-2
-3
(a)
193
-2
-1
0
1
θ
2
3
-3
(b)
-2
-1
0
1
2
3
θ
Fig. 17. Phase portraits for the planar oscillator (57): (a) typical case, (mz ; m3 ; t) = (−0:52; −0:76; 0:8); and (b) special
case for mz = m3 = 0 and m2z =f ∈ (0; 4) with a pair of homoclinic orbits, (mz ; m3 ; t) = (−0:6; −0:6; 0:8).
where m3 = m · d3 (details can be found in Reference [22]). The equation for is the well-known
equation for the inclination angle of the Lagrange top. Eq. (58) for
can be solved once is
known. This system of equations is invariant under the transformation s → −s, → , → − ,
→ − , and is therefore reversible.
Phase portraits of (57) have been studied in Reference [56]. By reversibility they have re6ection
symmetry about the axis. In addition they are symmetric about the vertical axis. The generic
phase portrait is depicted in Fig. 17(a). It has two sets of closed level curves of the Hamiltonian
H = 2 =2 + V () around two 5xed points. (These 5xed points correspond to helical solutions.)
Degenerate cases occur for special values of the parameters mz , m3 and f: for mz = m3 = 0 (57)
reduces to the equation for the planar elastica, for mz = m3 = 0 and m2z =f ∈ (0; 4) we obtain the
phase portrait with homoclinic orbits shown in Fig. 17(b), while for mz = m3 = 0 and m2z =f ∈ (0; 4)
the phase portrait consists solely of closed level curves around the origin, much like those outside
the homoclinic orbit in Fig. 17(b). In the following we shall only deal with the generic case and
consider the closed level curve not crossing the vertical axis highlighted in Fig. 17(a). The proof
can easily be adapted to deal with a level curve encircling the origin (see also [26]).
For the boundary condition (13) implies that
(−1=2) = ±(1=2) (mod 2):
(59)
By symmetry we can restrict our attention to the positive sign, so that (13) further yields (since
= 0)
(−1=2) = (1=2) (mod 2):
(60)
In order to be compatible with our choice of co-ordinate frames in Section 2.1 and to ensure that
d3y (0) = 0 we have to take as initial condition (0) = 0, since (0) = 0 (mod ).
Now, on our level curve consider an orbit with initial conditions at O. To obtain a full solution
we have to integrate from this point, where s = 0, backward to s = −1=2 (point A, say) and forward
to s = 1=2 (point B, say). Let point C be such that sC = Tp =2, where Tp is the period of the orbit,
194
and let D and E be the re6ections of C and A about the axis. S and T are the intersection points
of the level curve with the axis and lie in the 5xed point set of the reversing transformation.
We now apply the boundary conditions (59) and (60). There are two cases:
(i) The orbit under consideration has exactly period Tp =1. Then the points A, B, and C coincide,
and we shall show below that the boundary conditions imply that the centreline of the rod
forms a closed curve. Hence the solution is congruent with the solution that has s = 0 at S
(just ‘slide the clamp along the rod’), and is thus reversible.
(ii) By (59), points A and B must have the same and hence B coincides with E. But since the
arclength from A to C, sAC , is equal to the arclength from C to B, sCB , we have sAC = sCE .
Further, we have sCE = sCD + sDE , and since reversibility of (57) implies that sAC = sDE we
conclude that sCD = 0. Hence C, T and D coincide, and by the de5nition of C this means
that O coincides with S, i.e., the starting point must lie on the symmetry axis, making the
solution reversible.
It remains to show that Tp = 1 implies a closed rod (case (i)). To see this, form the vector product
of (9) with d3 and use (56) and (58) to obtain
m3 − mz cos sin + cos ;
(61)
fx + m0y =
sin m3 − mz cos cos − sin :
− fy + m0x =
(62)
sin Then
1
x(1=2) − x(−1=2) = { (1=2) − (−1=2)} cos (1=2);
(63)
f
1
y(1=2) − y(−1=2) = { (1=2) − (−1=2)} sin (1=2);
(64)
f
and since A and B coincide (hence (1=2) = (−1=2)) we 5nd that the x and y co-ordinates at
both end points are equal. The second clamp condition (14) then gives that = 0, i.e., the rod has
a closed centreline.
These symmetric solutions with the clamps at a non-symmetric position are not found in our numerical work. In our shooting method we use one of the clamps as a reference position and therefore
only solutions with symmetrically placed clamps are obtained. Closed solutions with non-symmetric
clamps lie on straight lines (d = 1) in the d − t diagram, as on the line connecting the two
5gures-of-eight in Fig. 2a.
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Instability and self-contact phenomena in the writhing of clamped rods

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